Carbon dating is accurate up to roughly 50,000 years ago.
After this the rate of decay is too small to get accurate dates from a sample, and other dating methods (such as Thermoluminescence and Potassium-Argon) have to be used. When something dies its body stops taking in carbon.
Animals take up carbon-14 when they eat plants or other animals.
These organisms no longer take in new carbon-14 once they've died, so the carbon-14 isn't replenished as it decays back into nitrogen.
And say that every 5,568 years you add another gallon of water to the mixture.
Doing that basically cuts the concentration of blue dye in half.
Plants take up carbon-14 when they convert CO2 into sugar and build cellular structures.
You then take a gallon of that diluted mixture and add another gallon of pure water to it 5,568 years later.
The concentration of blue dye is cut in half again.
Measure the number of decay events, and then compare the number of decays per unit mass per unit time to the decay activity in living organisms.
Carbon-14 dating has a 95 percent accuracy when determining the age of organic material up to 50,000 years old.
Now imagine repeating this process for quite some time.